Representation categories of Mackey Lie algebras as universal monoidal categories

TL;DR

This paper constructs and analyzes a universal monoidal category generated by two objects with a filtration, realized as a representation category of a Lie algebra, and describes its simple and injective objects, proving it is Koszul.

Contribution

It introduces a universal monoidal category associated with Mackey Lie algebras, explicitly describes its simple and injective objects, and proves its Koszul property.

Findings

Explicit description of simple objects in $ ext{T}_ ext{alpha}$

Explicit description of injective objects in $ ext{T}_ ext{alpha}$

Proof that $ ext{T}_ ext{alpha}$ is Koszul

Abstract

Let $\mathbb{K}$ be an algebraically closed field of characteristic $0$. We study a monoidal category $\mathbb{T}_\alpha$ which is universal among all symmetric $\mathbb{K}$-linear monoidal categories generated by two objects $A$ and $B$ such that $A$ has a, possibly transfinite, filtration. We construct $\mathbb{T}_\alpha$ as a category of representations of the Lie algebra $\mathfrak{gl}^M(V_*,V)$ consisting of endomorphisms of a fixed diagonalizable pairing $V_*\otimes V\to \mathbb{K}$ of vector spaces $V_*$ and $V$ of dimension $\alpha$. Here $\alpha$ is an arbitrary cardinal number. We describe explicitly the simple and the injective objects of $\mathbb{T}_\alpha$ and prove that the category $\mathbb{T}_\alpha$ is Koszul. We pay special attention to the case where the filtration on $A$ is finite. In this case $\alpha=\aleph_t$ for $t\in\mathbb{Z}_{\geq 0}$.